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Polygon Interior & Exterior Angle Sums

Every polygon's interior angles add up to a total you can predict from one number: how many sides it has. A triangle's angles total 180∘180^\circ, a quadrilateral's total 360∘360^\circ, a pentagon's total 540∘540^\circ — the pattern is that each extra side adds another 180∘180^\circ. The formula that captures this is S=(n−2)⋅180∘S = (n - 2) \cdot 180^\circ, where nn is the number of sides.

Exterior angles are even simpler. Take one exterior angle at each vertex of any convex polygon and they always add to exactly 360∘360^\circ — no matter how many sides the polygon has. These two facts, together, answer almost every polygon angle question you will see.

The interior angle sum formula

Pick one vertex of a polygon and draw every diagonal from it. The polygon splits into triangles — always n−2n - 2 of them for an nn-sided polygon. A hexagon splits into 44 triangles, an octagon into 66. Since each triangle contributes 180∘180^\circ, the interior angle sum is S=(n−2)⋅180∘S = (n - 2) \cdot 180^\circ.

In the figure, the diagonals from one vertex of a seven-sided polygon cut it into 7−2=57 - 2 = 5 triangles — the same n−2n - 2 pattern works for any convex polygon.

If the polygon is regular — all sides and all angles congruent — you can go one step further. Divide the sum by the number of angles to get each one: each interior angle of a regular polygon measures (n−2)⋅180∘n\dfrac{(n - 2) \cdot 180^\circ}{n}.

The exterior angle sum is always 360∘360^\circ

An exterior angle sits between one side of the polygon and the extension of the next side. Here is the picture to keep in mind: imagine walking all the way around the polygon along its edges. At each corner you turn through the exterior angle, and by the time you get back to the start you have made exactly one full turn. That is why the exterior angles of any convex polygon total 360∘360^\circ — for a triangle, a decagon, or a 100-gon.

At every vertex, the interior and exterior angles sit on a straight line, so they add to 180∘180^\circ. That link lets you switch between the two whenever one is easier to work with. For a regular polygon, each exterior angle is 360∘n\dfrac{360^\circ}{n}.

Working backwards to find the number of sides

A very common question gives you each interior angle of a regular polygon and asks how many sides it has. The fastest route runs through the exterior angle: subtract the interior angle from 180∘180^\circ to get the exterior angle, then divide it into 360∘360^\circ. In symbols, n=360∘exterior anglen = \dfrac{360^\circ}{\text{exterior angle}}.

You can also set up (n−2)⋅180n\dfrac{(n - 2) \cdot 180}{n} equal to the given angle and solve for nn, but the exterior angle route gets the same answer with far less algebra.

Worked examples

Example 1: interior angle sum of a hexagon

Find the sum of the interior angle measures of a hexagon.

A hexagon has 66 sides, so use the formula with n=6n = 6S=(n−2)⋅180∘S = (n - 2) \cdot 180^\circ
SubstituteS=(6−2)⋅180∘S = (6 - 2) \cdot 180^\circ
MultiplyS=4⋅180∘=720∘S = 4 \cdot 180^\circ = 720^\circ

Answer: 720∘720^\circ

Example 2: find the number of sides

Each interior angle of a regular polygon measures 144∘144^\circ. How many sides does the polygon have?

Find one exterior angle first — interior and exterior add to 180∘180^\circ180∘−144∘=36∘180^\circ - 144^\circ = 36^\circ
The exterior angles of any convex polygon total 360∘360^\circn⋅36∘=360∘n \cdot 36^\circ = 360^\circ
Dividen=36036=10n = \dfrac{360}{36} = 10

Answer: 1010 sides

Example 3: a missing interior angle

Four angles of a pentagon measure 100∘100^\circ, 110∘110^\circ, 95∘95^\circ, and 120∘120^\circ. Find the fifth angle.

Find the total for a pentagon, n=5n = 5S=(5−2)⋅180∘=540∘S = (5 - 2) \cdot 180^\circ = 540^\circ
Add the known angles100+110+95+120=425100 + 110 + 95 + 120 = 425
Subtract from the total540−425=115540 - 425 = 115

Answer: 115∘115^\circ

Try one yourself

Common questions

Does the exterior angle sum change when the polygon has more sides?

No. One trip around any convex polygon is one full turn, so the exterior angles always total 360∘360^\circ. More sides just means more, smaller turns. Only the interior angle sum grows with nn.

Do the formulas work for irregular polygons?

The sum formulas do — (n−2)⋅180∘(n - 2) \cdot 180^\circ for interior angles and 360∘360^\circ for exterior angles hold for any convex polygon. Dividing by nn to find each individual angle only works when the polygon is regular, because that step assumes every angle is equal.

Why is it n−2n - 2 triangles and not nn?

When you draw diagonals from a single vertex, that vertex and its two neighbors do not get their own triangle fanning out — the fan starts one vertex over and ends one vertex early. Count it on a hexagon: diagonals from one vertex make exactly 44 triangles, which is 6−26 - 2.

What is the fastest way to find each exterior angle of a regular polygon?

Divide: each exterior angle is 360∘n\dfrac{360^\circ}{n}. A regular pentagon has exterior angles of 3605=72∘\dfrac{360}{5} = 72^\circ, so each interior angle is 180−72=108∘180 - 72 = 108^\circ.

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